Exercise 4 Page 108

We are asked to solve the given quadratic equation. We will solve it by graphing. To solve a quadratic equation by graphing, we have to follow three steps.

Write the equation in standard form, ax^2+bx+c=0.
Graph the related function f(x)=ax^2+bx+c.
Find the x-intercepts, if any.

The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Let's write our equation in standard form. This means gathering all of the terms on the left-hand side of the equation. x^2+12=- 8x ⇔ x^2+8x+12=0 Now we can identify the function related to the equation.

Equation:& x^2+8x+12=0 Related Function:& f(x)=x^2+8x+12

Graphing the Related Function

To draw the graph of the related quadratic function written in standard form, we must start by identifying the values of a, b, and c. f(x)=x^2+8x+12 ⇕ f(x)= 1x^2+ 8x+ 12 We can see that a= 1, b= 8, and c= 12. Now, we will follow three steps to graph the function.

Find the axis of symmetry.
Make a table of values using x-values around the axis of symmetry.
Plot and connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation x=- b2 a. Since we already know the values of a and b, we can substitute them into the formula.

x=- b/2a

SubstituteII

a= 1, b= 8

x=- 8/2( 1)

▼

Evaluate right-hand side

Multiply

x=-8/2

CalcQuot

Calculate quotient

x=- 4

The axis of symmetry of the parabola is the vertical line with equation x=- 4.

Making the Table of Values

Next, we will make a table of values using x-values around the axis of symmetry x=- 4.

x	x^2+8x+12	f(x)
- 8	( - 8)^2+8( - 8)+12	12
- 6	( - 6)^2+8( - 6)+12	0
- 4	( - 4)^2+8( - 4)+12	- 4
- 2	( - 2)^2+8( - 2)+12	0
0	0^2+8( 0)+12	12

Plotting and Connecting the Points

We can finally draw the graph of the function. Since a=1, which is positive, the parabola will open upwards. Let's connect the points with a smooth curve.

Finding the x-intercepts

Let's identify the x-intercepts of the graph of the related function.

We can see that the parabola intersects the x-axis twice. The points of intersection are ( - 6,0) and ( - 2,0). Therefore, the equation x^2+8x+12=0 has two solutions, x= - 6 and x= - 2.